The apparent contact angle for a nearly spherical drop on a heterogeneous surface

For a wide class of heterogeneous surfaces, sessile drops in equilibrium form approximately spherical caps and a single apparent contact angle may be defined. This apparent contact angle is shown to be given by a derivative Cassie equation, in which the area fraction weightings of the Cassie equation are replaced by fractions involving the derivatives of areas with respect to changing radius. The derivative Cassie equation reduces to the classical Cassie equation for approximately uniform surfaces. An example is given where the correct contact angle is given by the derivative, but not the classical, Cassie equation.